Given The Following Vectors A ⃗ = ( 2 , 0 , 1 \vec{a} = (2, 0, 1 A = ( 2 , 0 , 1 ] And B ⃗ = ( 1 , 2 , 1 \vec{b} = (1, 2, 1 B = ( 1 , 2 , 1 ], Calculate − 2 A ⃗ + 3 B ⃗ -2 \vec{a} + 3 \vec{b} − 2 A + 3 B .A. − 2 A ⃗ + 3 B ⃗ = ( 0 , 0 , 0 -2 \vec{a} + 3 \vec{b} = (0, 0, 0 − 2 A + 3 B = ( 0 , 0 , 0 ]B. − 2 A ⃗ + 3 B ⃗ = ( − 1 , 6 , 1 -2 \vec{a} + 3 \vec{b} = (-1, 6, 1 − 2 A + 3 B = ( − 1 , 6 , 1 ]C. $-2

Introduction

Vectors are an essential concept in mathematics, particularly in linear algebra and geometry. They are used to represent quantities with both magnitude and direction. In this article, we will explore the concept of vector operations, specifically the calculation of $-2 \vec{a} + 3 \vec{b}$, where $\vec{a} = (2, 0, 1)$ and $\vec{b} = (1, 2, 1)$.

Understanding Vectors

A vector is a mathematical object that has both magnitude (length) and direction. It can be represented graphically as an arrow in a coordinate system. Vectors can be added, subtracted, and scaled, which are essential operations in vector calculus.

Vector Addition and Subtraction

Vector addition and subtraction are performed component-wise. For two vectors $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$, their sum is defined as:

$$\vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)$$

Similarly, the difference between two vectors is defined as:

$$\vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2, a_3 - b_3)$$

Scalar Multiplication

Scalar multiplication is a way of scaling a vector by a scalar value. For a vector $\vec{a} = (a_1, a_2, a_3)$ and a scalar $c$, the scaled vector is defined as:

$$c\vec{a} = (ca_1, ca_2, ca_3)$$

**Calculating $-2 \vec{a} + 3 \vec{b}$

Now that we have a basic understanding of vector operations, let's calculate $-2 \vec{a} + 3 \vec{b}$.

Given $\vec{a} = (2, 0, 1)$ and $\vec{b} = (1, 2, 1)$, we can calculate $-2 \vec{a}$ and $3 \vec{b}$ separately.

$$-2 \vec{a} = (-2 \cdot 2, -2 \cdot 0, -2 \cdot 1) = (-4, 0, -2)$$

$$3 \vec{b} = (3 \cdot 1, 3 \cdot 2, 3 \cdot 1) = (3, 6, 3)$$

Now, we can add the two scaled vectors:

$$-2 \vec{a} + 3 \vec{b} = (-4, 0, -2) + (3, 6, 3) = (-4 + 3, 0 + 6, -2 + 3) = (-1, 6, 1)$$

Conclusion

In this article, we have explored the concept of vector operations, specifically the calculation of $-2 \vec{a} + 3 \vec{b}$. We have seen how to perform vector addition, subtraction, and scalar multiplication, and how to apply these operations to calculate the desired result. The correct answer is:

$-2 \vec{a} + 3 \vec{b} = (-1, 6, 1)$

Discussion

Do you have any questions or doubts about vector operations? Have you encountered any challenges while working with vectors? Share your thoughts and experiences in the comments below!

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Vector Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Related Topics

• Vector addition and subtraction
• Scalar multiplication
• Dot product and cross product
• Linear transformations and matrices

Further Reading

• "Vector Calculus" by Michael Spivak
• "Linear Algebra and Its Applications" by Gilbert Strang
• "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

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Introduction

In our previous article, we explored the concept of vector operations, specifically the calculation of $-2 \vec{a} + 3 \vec{b}$. We have seen how to perform vector addition, subtraction, and scalar multiplication, and how to apply these operations to calculate the desired result. In this article, we will provide a Q&A guide to help you better understand vector operations and how to apply them in different scenarios.

Q&A

Q: What is the difference between vector addition and scalar multiplication?

A: Vector addition is the process of combining two or more vectors to form a new vector. Scalar multiplication, on the other hand, is the process of scaling a vector by a scalar value.

Q: How do you add two vectors?

A: To add two vectors, you simply add their corresponding components. For example, if we have two vectors $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$, their sum is defined as:

$$\vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)$$

Q: How do you subtract two vectors?

A: To subtract two vectors, you simply subtract their corresponding components. For example, if we have two vectors $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$, their difference is defined as:

$$\vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2, a_3 - b_3)$$

Q: What is the result of multiplying a vector by a scalar?

A: When you multiply a vector by a scalar, you are scaling the vector by that scalar value. For example, if we have a vector $\vec{a} = (a_1, a_2, a_3)$ and a scalar $c$, the scaled vector is defined as:

$$c\vec{a} = (ca_1, ca_2, ca_3)$$

Q: How do you calculate the magnitude of a vector?

A: The magnitude of a vector is the length of the vector. It can be calculated using the formula:

$$\|\vec{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2}$$

Q: How do you calculate the dot product of two vectors?

A: The dot product of two vectors is a scalar value that represents the amount of "similarity" between the two vectors. It can be calculated using the formula:

$$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$$

Q: How do you calculate the cross product of two vectors?

A: The cross product of two vectors is a vector that is perpendicular to both of the original vectors. It can be calculated using the formula:

$$\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$$

Conclusion

In this article, we have provided a Q&A guide to help you better understand vector operations and how to apply them in different scenarios. We have covered topics such as vector addition, subtraction, scalar multiplication, magnitude, dot product, and cross product. We hope that this guide has been helpful in clarifying any doubts you may have had about vector operations.

Discussion

Do you have any questions or doubts about vector operations? Have you encountered any challenges while working with vectors? Share your thoughts and experiences in the comments below!

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Vector Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Related Topics

• Vector addition and subtraction
• Scalar multiplication
• Dot product and cross product
• Linear transformations and matrices

Further Reading

• "Vector Calculus" by Michael Spivak
• "Linear Algebra and Its Applications" by Gilbert Strang
• "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Note: The references and related topics are provided for further reading and exploration. They are not directly related to the Q&A guide.